Generalized Discrete Spherical Harmonic Transforms

Two generalizations of the spherical harmonic transforms are provided. First, they are generalized to an arbitrary distribution of latitudinal points θi. This unifies transforms for Gaussian and equally spaced distributions and provides transforms for other distributions commonly used to model geophysical phenomena. The discrete associated Legendre functions Pnm(θi) are shown to be orthogonal, to within roundoff error, with respect to a weighted inner product, thus providing the forward transform to spectral space. Second, the representation of the transforms is also generalized to rotations of the discrete basis set Pnm(θi. A discrete function basis is defined that provides an alternative to Pnm(θi. On a grid with N latitudes, the new basis requires O(N2) memory compared to the usual O(N3). The resulting transforms differ in spectral space but provide identical results for certain applications. For example, a forward transform followed immediately by a backward transform projects the original discrete function in a manner identical to the existing transforms. Namely, they both project the original function onto the same smooth least squares approximation without the high frequencies induced by the closeness of the points in the neighborhood of the poles. Finally, a faster projection is developed based on the new transforms.